ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH

Given a finite group G, the generating graph 0.G/ of G has as vertices the (nontrivial) elements of G and two vertices are adjacent if and only if they are distinct and generate G as group elements. In this paper we investigate properties about the degrees of the vertices of 0.G/ when G is an alternating group or a symmetric group of degree n. In particular, we determine the vertices of 0.G/ having even degree and show that 0.G/ is Eulerian if and only if n > 3 and n and n 1 are not equal to a prime number congruent to 3 modulo 4.